Ex.1, P.354

A die is rolled 24 times. Use the Central Limit Theorem to estimate the probability that

  1. the sum is greater than 84.

P(S24 > 84) - ?

The sum is a random variable: S(24) = X1 + X2 + … + X24,

where each Xj has distribution

Mx = 1(1/6)+2(1/6)+3(1/6)+4(1/6)+5(1/6)+6(1/6)= 7/2

σ2 = V(X)=16(25/4+9/4+1/4+1/4+9/4+25/4)=35/12

E(S24) = 24 * 7/2 = 84

σ2(S24) = 24 * 35/12 = 70

σ = sqrt(70) = 8.367

P(S24) = (84.5−84)/8.366

# 
P = (84.5 - 24 * 7/2)/sqrt(24 * 35/12)
p_h<- 1 - pnorm(P)
p_h
## [1] 0.4761728
  1. the sum is equal to 84.

P(S24 = 84) - ?

P = (83.5 - 24 * 7/2)/sqrt(24 * 35/12)
p_l <- pnorm(P)
p_l
## [1] 0.4761728

P(S24 = 84) = 1- (P(S24 > 84) + P(S24 < 84))

1- (p_h+p_l)
## [1] 0.04765436